(D) Given that $A$ is a skew-symmetric matrix,we have $A^T = -A$.Let $M = XA + AX^T$. We want to find the determinant $|M|$.Taking the transpose of $M

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(D) Given that $A$ is a skew-symmetric matrix,we have $A^T = -A$.Let $M = XA + AX^T$. We want to find the determinant $|M|$.Taking the transpose of $M$:$M^T = (XA + AX^T)^T = (XA)^T + (AX^T)^T = A^T X^T + (X^T)^T A^T = A^T X^T + X A^T$.Substituting $A^T = -A$:$M^T = (-A) X^T + X (-A) = -AX^T - XA = -(XA + AX^T) = -M$.Since $M$ is a matrix of order $3$,we have $|M^T| = | -M | = (-1)^3 |M| = -|M|$.Thus,$|M| = -|M|$,which implies $2|M| = 0$,so $|M| = 0$.

Class 12 Mathematics 3-4. Matrices and Determinants Mix Examples-Determinants and Matrices

If $A$ is a skew-symmetric matrix of order $3$ and $X$ is another matrix of the same order,then $|XA + AX^T|$ is (where $|P|$ denotes the determinant of matrix $P$).

A
$|X + X^T|$
B
$|A + X|$
C
$|A - X|$
D
$0$

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If $A$ is a skew-symmetric matrix of order $3$ and $X$ is another matrix of the same order,then $|XA + AX^T|$ is (where $|P|$ denotes the determinant of matrix $P$).

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